3x^2-19x-31=0

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Solution for 3x^2-19x-31=0 equation: 


Simplifying
3x2 + -19x + -31 = 0

Reorder the terms:
-31 + -19x + 3x2 = 0

Solving
-31 + -19x + 3x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-10.33333333 + -6.333333333x + x2 = 0

Move the constant term to the right:

Add '10.33333333' to each side of the equation.
-10.33333333 + -6.333333333x + 10.33333333 + x2 = 0 + 10.33333333

Reorder the terms:
-10.33333333 + 10.33333333 + -6.333333333x + x2 = 0 + 10.33333333

Combine like terms: -10.33333333 + 10.33333333 = 0.00000000
0.00000000 + -6.333333333x + x2 = 0 + 10.33333333
-6.333333333x + x2 = 0 + 10.33333333

Combine like terms: 0 + 10.33333333 = 10.33333333
-6.333333333x + x2 = 10.33333333

The x term is -6.333333333x.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333x + 10.02777778 + x2 = 10.33333333 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333x + x2 = 10.33333333 + 10.02777778

Combine like terms: 10.33333333 + 10.02777778 = 20.36111111
10.02777778 + -6.333333333x + x2 = 20.36111111

Factor a perfect square on the left side:
(x + -3.166666667)(x + -3.166666667) = 20.36111111

Calculate the square root of the right side: 4.51232879

Break this problem into two subproblems by setting 
(x + -3.166666667) equal to 4.51232879 and -4.51232879.


Subproblem 1
x + -3.166666667 = 4.51232879

Simplifying
x + -3.166666667 = 4.51232879

Reorder the terms:
-3.166666667 + x = 4.51232879

Solving
-3.166666667 + x = 4.51232879

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + x = 4.51232879 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + x = 4.51232879 + 3.166666667
x = 4.51232879 + 3.166666667

Combine like terms: 4.51232879 + 3.166666667 = 7.678995457
x = 7.678995457

Simplifying
x = 7.678995457


Subproblem 2
x + -3.166666667 = -4.51232879

Simplifying
x + -3.166666667 = -4.51232879

Reorder the terms:
-3.166666667 + x = -4.51232879

Solving
-3.166666667 + x = -4.51232879

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + x = -4.51232879 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + x = -4.51232879 + 3.166666667
x = -4.51232879 + 3.166666667

Combine like terms: -4.51232879 + 3.166666667 = -1.345662123
x = -1.345662123

Simplifying
x = -1.345662123


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.678995457, -1.345662123}

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